All the while, the airline remained profitable and has seen a long-term upward trend in its stock. In fact, it was recently rated as one of the top places to work in America by jobs posting and review platform Glassdoor.

So what's Kelly's secret?

"There is a lot to a culture, and it's easier to have a strong culture if you feel like you're a champion," he says. "And that's the way our employees feel."

Many Southwest employees agree with the CEO.

In anonymous job reviews on Glassdoor, dozens of employees cited the same reasons why the company is a great place to work.

1. A culture of appreciation

Employees say that managers at Southwest make them feel valued.

"Every leader I've met has always made me feel like my work was appreciated and made a difference," one customer support specialist writes.

2. Team unity

Kelly says he encourages every team to treat each other like family. Apparently, the strategy has been working.

"[The] culture is awesome! The company really takes care of their employees and understands their needs," a project manager writes.

3. Opportunities for advancement

Southwest provides plenty of opportunities for employees to either advance or move around within the company, dozens of employee reviews show.

"They take their time with their hiring process and it works, because every leader I've met has always made me feel like my work was appreciated and made a difference," one employee writes.

"It's just a great environment to work and live in," another employee writes. "I don't see myself working for another company."

4. Great benefits

Good pay and a variety of benefits don't hurt, either.

As one ramp agent puts it, the company has "great work schedule flexibility in most departments within Ground Operations and Inflight."

Free unlimited flights within company destinations, a competitive 401(k) company match and a great stock purchase plan are other perks employees appreciate.

Steve Proehl

Of course, not every employee is happy with the company. Earlier this year, hundreds of Southwest workers protested outside of a shareholder meeting. Some employees say they are "required to work unreasonably long hours of overtime" when the company is short-staffed, according to employee reviews on Glassdoor. Others say that the pay is low.

But Southwest's leadership says it is committed to treating its workers "like family."

Glassdoor Community Expert Scott Dobroski, who has personally read hundreds of the company's reviews, says there does seem to be something special about the airline.

"What's happening at Southwest is exactly what we like to call 'the Southwest effect,'" Dobroski tells CNBC. "Happy employees mean happy customers, which in turn has a direct tie to a strong financial performance."